Effect of the Accuracy of Target Range Measurement on the Accuracy of Fire

AiMT Advances in Military Technology

Vol. 5, No. 2, December 2010

Effect of the Accuracy of Target Range

Measurement on the Accuracy of Fire

R. Vítek*

and L. Jedlička

Department of Weapons and Ammunition, University of Defence, 662 10, Brno, Czech Republic

The manuscript was received on 10 June 2010 and was accepted after revision for publication on

14 September 2010.

Abstract:

The article deals with the mutual relation between the accuracy of the target range

measurement and the accuracy of fire. The stereoscopic target range measurement and

its errors are briefly described in the first part. The second part of the article is focused

on the impact of inaccurate target range measurement on projectile’s vertical position of

the point of impact. Various ballistic systems are used for the simulation of the effect.

Results of simulations are discussed from two points of view. First point of view is the

use of the results for the selection of the most suitable ballistic system for the firing on

the target at given range. The second one offers utilisation of the results for the setting of

requirements on the accuracy of target range measurement for given ballistic system and

size of the target.

Keywords:

Exterior ballistics, passive range finder, target range measurement, trajectory modelling

1. Introduction

The fire accuracy of the guns is largely affected by the accuracy of the determination

of the relative position between the weapon and the target. In case of the direct fire,

the most important characteristic of the target position with respect to the weapon is

the slant range as the basic input factor for the fire data computation [1].

Nowadays, the laser rangefinder is the most frequently used device for the target

slant range measurement for its promptness, accuracy and reliability. Nevertheless, the

recent research [2, 3] shows the relatively high rate of the laser measurement errors

due to the atmosphere effects, false reflections, laser beam divergence, and others.

From the tactical point of view, the most important disadvantage of the laser

* Corresponding author: Department of Weapons and Ammunition, University of Defence, Kounicova 65,

CZ-662 10 Brno, Czech Republic, phone: +420 973 445 165, e-mail: [email protected]

70 R. Vítek and L. Jedlička

rangefinder is the fact that the laser rangefinder during its operation emits a relatively

large amount of energy, which is detectable by the enemy‟s reconnaissance means [1].

Especially in the situations when it is necessary to measure the target range several

times (repeating of the measurement because of the measurement fault, effort to

determine the direction and velocity of the target motion by means of the several

measured points of the target trajectory and so on) the enemy gets sufficient time and

information for taking the appropriate countermeasures, or the counterattack.

One possible solution of this problem is to utilize the methods of the digital

photogrammetry when the target position and the slant range are determined from at

least two images of the target which were synchronously acquired from the different

camera positions. The fundamental advantage of this approach is that the CCD or

CMOS detectors of cameras do not emit such an amount of energy which could be

detected by the enemy, and only the radiation emitted or reflected by the target is

passively received. Furthermore, these types of detectors could operate in the

relatively wide range of the electromagnetic spectra, so they can be used for target

range measurement also under reduced visibility and in the night.

2. Passive Target Range Measurement

The determination of the target position based on the image information processing

results from the fundamental relations between the target coordinates with respect to

the global coordinate system X, Y, and Z connected with the Earth and the image

coordinates of the target with respect to the local coordinate system ys and zs,

connected with the camera detector. This relation can be expressed by the formulas in

following form [4, 5]

11 0 21 0 31 0

0

13 0 23 0 33 0

12 0 22 0 32 0

0

13 0 23 0 33 0

,

,

s

s

r X X r Y Y r Z Zz z c

r X X r Y Y r Z Z

r X X r Y Y r Z Zy y c

r X X r Y Y r Z Z

(1)

where:

ys, zs are the object image coordinates with respect to the local coordinate

system connected with the camera detector,

y0, z0 are the coordinates of the principal point of the camera with respect to the

local coordinate system connected with the camera detector,

c is the principal distance of the camera,

X0, Y0, Z0 are the coordinates of the projection centre of the camera with respect

to the global coordinate system,

X, Y, Z are the coordinates of the target with respect to the global coordinate

system and

rij are the elements of the rotational transformation matrix of the camera with

respect to the global coordinate system.

The slant range L of the target can be determined using the following formula

2 2 2

0 0 0L X X Y Y Z Z . (2)

The principal point coordinates and the camera constant are the elements of the

inner camera orientation. These elements are determined during camera calibration

71 71

Effect of the Accuracy of Target Range Measurement

on the Accuracy of Fire

and they are considered to be the constants for the measurement process. The

projection centre coordinates and the rotation of the camera with respect to the global

coordinate system are the elements of the outer camera orientation and they can vary

during the measurement process in general. However, in the scope of the slant range

determination the relative elements of the outer camera orientation are substantial, i.e.

the knowledge of the relative position of the reference camera with respect to the

measuring camera is more important. The elements of the relative outer orientation of

the camera are also determined during camera calibration and they also remain

constant.

In the case of incorrect determination of the inner and relative outer camera

orientation elements, systematic error is introduced into the measurement process.

This error can be eliminated by re-calibration of the camera, though.

Random error in the slant range measurement process is caused by errors in the

determination of the image coordinates of the target. Hereinafter the effect of the

incorrect image coordinates determination on the slant range computation and

consequently on the target hit error will be discussed.

It is obvious from the formulas (1) that the image coordinates of the target can be

determined from one image. But for the inverse operation, when the global coordinates

of the target shall be reconstructed for the known image coordinates, one image is

insufficient, because we have only two conditional equations for three unknown

coordinates X, Y, Z. Therefore, at least two images have to be acquired for obtaining

four conditional equations, which can be used for deriving the equation system with

the unique solution. In the case of using two cameras the term camera base is used.

The camera base b is the element of the relative outer camera orientation in principle

and it can be expressed by formula

2 2 2

01 00 01 00 01 00b X X Y Y Z Z , (3)

where:

X00, Y00, and Z00 are the coordinates of the reference camera projection centre

with respect to the global coordinate system and

X01, Y01, and Z01 are the coordinates of the measuring camera projection centre

with respect to the global coordinate system.

For better understanding, the following conditions have been adopted:

the optical axes of the both cameras are parallel to each other and perpendicular

to the base of the cameras,

the beginning of the global coordinate system is identical with the projection

centre of the reference camera,

the optical axis of the reference camera is coincidental with the longitudinal

axis X of the global coordinate system and it goes through the point which

represents the target,

the axes y and z of the local coordinate system connected with the camera

detector are parallel to the corresponding axes Y and Z of the global coordinate

system and

for the measuring camera it is valid that X01 = X00, Y01 = Y00 and Z01 = b.

This normal case of stereo restitution is shown in Fig. 1. The individual symbols

in this figure will be explained in this article hereafter. On the basis of these

preliminary conditions we can transform the relation between the target global

coordinates and image coordinates into the simplified form


1 01 1s

bz z c

L , (4)

where:

zs1 is the horizontal image coordinate of the target for the measuring camera,

z01 is the horizontal coordinate of the principal point of the measuring camera,

c1 is the principal distance of the measuring camera,

b is the base of the cameras and

L is the slant range of the target with respect to the projection centre of the

reference camera.

From equation (4) the resulting formula for the slant range computation can be

derived in form

1

1 01s

cL b

z z

. (5)

Fig. 1 Normal case of the stereo restitution

3. Errors of the Passive Target Range Measurement

In the case of using the digital cameras for acquiring the images, the fundamental error

in the image coordinates determination is caused by the spatial discretization of the

continuous brightness distribution in the plane of the camera detector. It means that the

73 73



image coordinate zs1 does not take the values from the continuous interval 0; mz ,

where zm is the physical width of the camera detector but from the vector of discrete

values Z whose elements can be computed by means of the formula

iz zi for i = 0..nm 1, (6)

where:

z is the minimal distinguishable spatial increment (or spatial resolution) of the

camera detector and

nm is the number of spatial increments in the particular direction.

In practice the z is frequently equal to the dimension of the camera detector element

(pixel).

If the determined value of the discrete image coordinate is 1sz , for this value

following formula is valid

1 1s zz zn , (7)

where:

nz1 is the number of the image pixel, which was estimated as the representation

of the target.

In general, due to the quantization noise mentioned above, the nominal value of

image coordinate zs1 will differ from 1sz and it can take its value from the interval

1 1 11 ; 1s z zz z n z n . (8)

To determine the error in the slant range measurement it is necessary to examine

the two limiting values of the image coordinate from the interval above.

Using the substitution

1 01 1s pz z zn , (9)

where np1 = nz1 – z01/z, the equation (5) can be written in form

1

1p

bcL

zn

. (10)

The slant range error ΔL is given by the difference between the evaluated slant

ranges for limiting values of the discrete image coordinates according to the formula

1 1

1 1

1 1

1 11 1

2

1 1 1

1

2

1

1 1

1 11 1

1 1 1

2.

1

p p

p p

p p p

p

bc bcL L L b

z n z n

n nbc bc

z n n z n

bc

z n

(11)

Using the formula (10), it can be written for np1

1

1p

bcn

zL

. (12)


Substituting (12) into (11), the new formula for the determination of the slant

range error will be obtained

22 2

1 1 1

2 2 2 2 2 2 2 2 2 2

1 1 1

2 2

22 2

1

c bc bc zLz LL b

z zb c b c z L b c z L

z L

. (13)

For relatively high resolutions of camera detectors (Δz < 10–5

m) and relatively

short target ranges (L < ca. 3000 m) it can be written

2 2 0z L , (14)

therefore, the formula (13) can be transformed into form

2

1

2 zLL

bc

. (15)

It means that absolute error of the measured slant range is indirectly

proportional to the base of cameras and principal distance of the camera and directly

proportional to the resolution of the camera detector and the square of the measured

distance. Finally, the relative error of the slant range measurement can be expressed by

equation in form

1

2L zL

L bc

. (16)

The estimated error of the range measurement can be used for determination of

the point of impact shift with respect to the aiming point.

Example: Δz = 6.45 μm, b = 2 m, c1 = 0.072 m, target range L varies from 300 to

1200 m. The estimated errors are shown in the Tab. 1.

Tab. 1 Estimated relative error of the measured target distance

Nominal target distance L [m] 300 600 900 1200

Image coordinates zs – z01 [mm] 0.480 0.240 0.160 0.120

Discretized image

coordinates Δznp1 [mm] 0.477 0.239 0.161 0.123

Discretized target distance

L [m] 301.70 603.39 893.02 1175.03

Relative difference between

L and L [%] 0.56 0.56 0.78 2.08

Estimated error ΔL [m] 8.15 32.61 71.44 123.69

Relative estimated error [%] 2.7 5.4 8.0 10.5

4. Ballistic Considerations

The current experimental firing task solved in the research project can be described as

a standard NATO target of size 2.3 2.3 m moving at distances up to 650 m with

velocity about 5 m·s1

.

At this moment the ballistic system of calibre 7.6254 mm for the purposes of

the research project is utilised. The used ballistic system is capable to successfully

75 75



fulfil the current experimental firing task, but it is not suitable for the accurate firing at

the longer ranges of fire which would be closer to the firing tasks in military practice,

especially due to increasing steepness of the projectile trajectory at longer ranges of

fire and also due to its sensitivity to atmospheric effects (especially cross wind).

For these reasons it is necessary to consider selection of a new ballistic system

that will be capable to successfully hit the moving standard target at ranges of fire

about 1000 m. The weapon system with this capability would be more suitable for the

application into practice.

4.1. Exterior Ballistic Modelling

The question of exterior ballistic modelling can be divided into two parts. The first

part is focused on the selection and use of a suitable trajectory model. The second part

deals with the most efficient calculation of the aiming angles.

Trajectory Model

Among several available trajectory models the point mass trajectory model was

chosen. This model is sufficiently accurate for the calculation of trajectories for the

direct fire ballistic systems and also the input data of various ballistic systems are

relatively easily available. The trajectories can be calculated very quickly in

comparison with the more sophisticated trajectory models and the speed of calculation

is a very important parameter in the ballistic computers.

This point mass trajectory model consists of six first order differential equations.

These equations describe projectile motion as a motion of a point mass in the space.

0.5

0

43 43

0

0.5

0

43 43

0

0.5

0

43 43

0

d,

d

d,

d

d,

d

d,

d

d,

d

d,

d

x N N

x x

N N

x

y N N

y y

N N

y

N Nz

z z

N N

z

v pc v w G

t p

xv

t

v pc v w G g

t p

yv

t

pvc v w G

t p

zv

t

(17)

where:

c43 is ballistic coefficient (drag law 1943),

pN is standard atmospheric pressure,

p0N is standard atmospheric pressure at sea level,

w is wind speed,

ηN is standard virtual air temperature,

η0N is standard virtual air temperature at sea level,

G43 is drag function (drag law 1943).


Details of the exterior ballistic model can be found in the [6, 7, 8, 9].

The system of equations (17) is solved by means of standard Runge-Kutta

method with variable time step.

Calculation of Aiming Angles

The basic function of any ballistic computer is to calculate the aiming angles (angle of

elevation and bearing) for the particular firing task. To successfully fulfil this task it is

necessary to know the following:

trajectory model (in this case the point mass trajectory model),

characteristics of weapon system (initial velocity, ballistic coefficient),

atmospheric conditions (air pressure, temperature, direction and speed of wind),

position of the target (slant range of the target, azimuth and angle of site).

The angle of elevation can be calculated for the particular firing task iteratively.

In this case one of the most efficient methods, the secant method, was used. The

method can be described by the following relation:

0 0

0 0

1 21 0 1

1 2

i ii i y i

y i y i

(18)

where:

θ0 is the angle of elevation,

y is the height of point of impact on the target.

The iteration process is ended when the height of point of impact y differs from

0 (centre of the target) by less than 1 calibre, i.e. 7.62·10–3

m. All calculations are

carried out under standard atmospheric conditions [10].

4.2. Selection of Ballistic Systems

For the purpose of this analysis, the most widespread and available ballistic systems

from the calibre 5.56 up to calibre 30 mm were chosen. It is necessary to mention that

the attention is primarily focused on the weapon systems of calibres from 12.7 up to

20 mm. Ballistic systems of calibre higher than 20 mm are too powerful for the

existing experimental gun-carriage and there are also other practical problems (e.g.

large danger areas in firing ranges, high cost of ammunition, …).

Among the small calibre ballistic systems were chosen following systems fielded

in the Army of the Czech Republic (ACR) or other NATO armies:

5.5645 mm (standard NATO),

7.6239 mm (ACR),


7.6254 mm (ACR),


12.7108 mm (ACR),

14.5114 mm (ACR).

Basic exterior ballistic properties of above mentioned small calibre ballistic systems

are summarised in Tab. 2.

77 77



Tab. 2 Exterior ballistic characteristics of selected small calibre ballistic systems

5.564

5

7.623

9

7.625

1

7.625

4

12.79

9

12.71

08

14.51

14 v0 [m·s

1]

1006 743 856 855 945 850 890

c43

[kg·m2

]

10.50 10.30 6.70 6.57 3.60 3.60 3.50

Xmax [m] 2853 2587 3833 3888 6345 6057 6309

Automatic cannons of following calibres were added rather due to the complexity

of the selection:

20102 mm (ACR),

30165 mm (ACR),

30211 mm (ACR).

Basic exterior ballistic properties of chosen automatic cannons are summarised in

Tab. 3.

Tab. 3 Summary of exterior ballistic characteristics of selected automatic cannons

20102 30165 30211

v0 [m·s1

]

1050 970 1000

c43 [kg·m2

] 4.55 2.05 2.00

Xmax [m] 5551 9756 10110

5. Effect of Accuracy of the Target Range Measurement

The range of the target is one of the most important characteristics of the target for the

calculation of the aiming angles. The accuracy of this characteristic directly affects the

accuracy of fire and consequently also the target hit probability.

The effect of the accuracy of the target range measurement on the position of the

point of impact is expressed by means of change of height of point of impact on

the target. Calculation of aiming angles is realised for the measured target range (with

error) and the projectile trajectory is calculated for the real (correct) target range and

the height of point of impact is recorded. The height of point of impact should be

considered in terms of the mean point of impact.

Using the formula (16) we can estimate that the range of errors of the target range

measurement will not exceed ±15 % of the target range up to 3500 m (for the

rangefinder configuration mentioned in the example in Chapter 3). The change of the

height of point of impact is presented in dependency on the relative range of the target.

The relative range of the target is defined as a ratio of both measured and real range of

the target. Three real ranges of the target (600, 900, and 1200 m) were chosen for the

calculations.

Results of these calculations are presented in the following figures and tables.

The tables show the results with the 5 % step. The currently used weapon systems are

marked dark grey and the perspective weapon systems are marked light grey.


Fig. 2 Effect of inaccuracy of target range measurement – 600 m

Tab. 4 Change of height of point of impact due to target range inaccuracy – 600 m

Error of target range measurement [m]

Calibre –15 % –10 % –5 % 0 % +5 % +10 % +15 %

5.5645 mm 0.89 0.62 0.32 0.00 0.35 0.73 1.14

7.6239 mm –1.81 –1.25 –0.64 0.00 0.68 1.39 2.14

7.6251 mm –0.79 –0.54 –0.27 0.00 0.29 0.59 0.91

7.6254 mm –0.77 –0.53 –0.27 0.00 0.28 0.58 0.89

12.799 mm –0.43 –0.29 –0.15 0.00 0.15 0.30 0.46

12.7108 mm –0.54 –0.36 –0.18 0.00 0.19 0.38 0.58

14.5114 mm –0.48 –0.33 –0.16 0.00 0.17 0.34 0.52

20102 mm –0.38 –0.26 –0.13 0.00 0.13 0.27 0.41

30165 mm –0.35 –0.23 –0.12 0.00 0.12 0.24 0.36

30211 mm –0.32 –0.22 –0.11 0.00 0.11 0.22 0.33

79 79






Calibre –15 % –10 % –5 % 0 % +5 % +10 % +15 %

5.5645 mm –4.04 –2.81 –1.47 0.00 1.58 3.29 5.12

7.6239 mm –6.40 –4.40 –2.26 0.00 2.40 4.95 7.64

7.6251 mm –2.85 –1.98 –1.03 0.00 1.11 2.30 3.57

7.6254 mm –2.78 –1.93 –1.00 0.00 1.08 2.24 3.47

12.799 mm –1.19 –0.81 –0.41 0.00 0.42 0.87 1.32

12.7108 mm –1.51 –1.03 –0.52 0.00 0.54 1.11 1.69

14.5114 mm –1.34 –0.91 –0.46 0.00 0.48 0.97 1.49

20102 mm –1.11 –0.75 –0.39 0.00 0.40 0.83 1.27

30165 mm –0.86 –0.58 –0.29 0.00 0.30 0.60 0.91

30211 mm –0.80 –0.54 –0.27 0.00 0.28 0.56 0.85





Calibre –15 % –10 % –5 % 0 % +5 % +10 % +15 %

5.5645 mm –11.30 –7.85 –4.09 0.00 4.44 9.25 14.45

7.6239 mm –16.24 –11.22 –5.82 0.00 6.26 12.99 20.24

7.6251 mm –7.71 –5.32 –2.75 0.00 2.94 6.07 9.39

7.6254 mm –7.52 –5.20 –2.69 0.00 2.87 5.93 9.17

12.799 mm –2.64 –1.81 –0.93 0.00 0.97 2.00 3.08

12.7108 mm –3.42 –2.34 –1.20 0.00 1.26 2.60 4.02

14.5114 mm –2.97 –2.03 –1.04 0.00 1.09 2.25 3.47

20102 mm –2.62 –1.81 –0.93 0.00 1.00 2.06 3.21

30165 mm –1.72 –1.16 –0.59 0.00 0.60 1.22 1.85

30211 mm –1.59 –1.07 –0.54 0.00 0.56 1.13 1.71

81 81



6. Discussion

The obtained results can be utilised in two different ways:

for selection of optimal ballistic system,

for setting of requirements on the accuracy of target range measurement.

6.1. Selection of Optimal Ballistic System

In this first case the range of fire, accuracy of the target range measurement, and the

size of the target are given and the optimal ballistic system is searched. A new firing

task is defined as a standard NATO target of size 2.3 x 2.3 m moving at ranges up to

1200 m with velocity of about 5 m·s–1

. The aiming point is placed into the centre of

the target.

The selection of the optimal ballistic system will be shown on the example of the

7.6254 and the 12.799 systems. From Tables 4, 5, and 6 the heights of point of

impact for all ranges of target and the ±5 % accuracy of measurement were separated

and placed into the Fig. 5. The horizontal lines represent the bottom and upper edge of

the target.

From Fig. 5 it can be clearly seen that the target range of 900 m is the limit for

the currently used ballistic system 7.6254; other effects (e.g. cross wind) are not

taken into account for this assessment. On the other hand, the perspective ballistic

system of calibre 12.799 can be used with the ±5 % accuracy of target range

measurement against targets at distances up to 1200 m. It should be also mentioned

that the marks in the figure represent the mean points of impact. The position of the

mean point of impact at the target edge (upper or bottom) means that only 50 % of all

projectiles fired will hit the target.

Fig. 5 Comparison of current and perspective ballistic system

+ - 7.6254

x - 12.799


6.2. Setting of Requirements on the Accuracy of Target Range Measurement

In this case the particular ballistic systems, range of fire and size of the target are

given and the required accuracy of the target range measurement is searched for. In

other words, the question is how accurately the target range must be measured for

a given ballistic system and the size of the target.

The setting of requirements on the accuracy of the measurement can be shown on

the example of the chosen ballistic systems and standard NATO target at the distance

of 1200 m as shown in Fig. 5. To hit the standard NATO target at this distance with

currently used ballistic system, it is necessary to measure the range of the target

with better than ± 2 % accuracy. In case of using the 12.799 system the target range

must be measured with better than ±6 % accuracy. For the 30 mm ballistic systems it

is sufficient yet the ±9 % accuracy of measurement.

Fig. 6 Comparison of current and perspective ballistic system

Figures 2 and 3 present the required accuracy of the target range measurement for

each ballistic system and given size of the target in a similar way.

7. Conclusion

The simple mathematical model of passive target range measurement is presented in

this article. This model was used for deriving the relative range error. From the error

analysis of the passive range measuring system, the minimal magnitudes of errors of

the target range measurement were obtained. This analysis became the basis for the

external ballistics assessment of the effect of the accuracy of the target range

measurement on the accuracy of fire.

83 83



From the ballistic analysis which has been conducted it can be concluded that

from the investigated ballistic systems the 30 mm automatic cannons show the

smallest change of the height of point impact. On the other hand, these ballistic

systems are too powerful and therefore they are not suitable for mounting on the

existing gun carriage. The size of the dangerous areas and cost of munition for these

systems cannot be neglected either.

From the perspective group of ballistic systems (12.7 – 20 mm) the ballistic

systems 12.799 and 20102 are the most suitable ones. These ballistic systems pose a

good and comparable insensitivity to the target range measurement accuracy but from

the performance point of view the 12.799 system is more suitable than the 20102

one.

The conclusion based on the analysis is that the ballistic system 12.799 should

be used for the firing against targets at distances about 1000 m. The dispersion of fire

of this ballistic system must be also investigated to get complete information for the

final decision.

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Acknowledgement

The work presented in this paper has been supported by the Ministry of Defence of the

Czech Republic (research project No. MO FVT0000402).

Documents

Effect of the Accuracy of Target Range Measurement on the Accuracy of Fire